To solve the problem, we need to analyze the function \( f(x) = [x \sin(\pi x)] \), where \([x]\) denotes the greatest integer less than or equal to \(x\). We will determine the continuity and differentiability of this function at various points.
### Step 1: Analyze the function \( f(x) = [x \sin(\pi x)] \)
1. **Understanding the components**:
- The function \( \sin(\pi x) \) oscillates between -1 and 1 for all \( x \).
- Therefore, \( x \sin(\pi x) \) will also oscillate, but its amplitude will depend on the value of \( x \).
### Step 2: Determine the behavior of \( f(x) \)
2. **Behavior at specific intervals**:
- For \( x \in [0, 1) \):
- \( \sin(\pi x) \) is non-negative and reaches its maximum at \( x = 1/2 \).
- Hence, \( 0 \leq x \sin(\pi x) < 1 \).
- Thus, \( f(x) = [x \sin(\pi x)] = 0 \) in this interval.
- For \( x \in [1, 2) \):
- \( \sin(\pi x) \) becomes negative after \( x = 1 \).
- Therefore, \( x \sin(\pi x) \) will be negative, and since \( x \) is positive, \( f(x) = [x \sin(\pi x)] \) will be less than or equal to -1.
- Hence, \( f(x) = -1 \) in this interval.
- For \( x \in [-1, 0) \):
- Here, \( \sin(\pi x) \) is also negative, and since \( x \) is negative, \( x \sin(\pi x) \) will be positive.
- Thus, \( f(x) = [x \sin(\pi x)] = 0 \).
- For \( x \in [-2, -1) \):
- Similar reasoning applies, and we find that \( f(x) = -1 \).
### Step 3: Check continuity and differentiability
3. **Continuity at \( x = 0 \)**:
- As \( x \) approaches 0 from the left and right, \( f(x) \) approaches 0. Thus, \( f(x) \) is continuous at \( x = 0 \).
4. **Continuity at \( x = 1 \)**:
- As \( x \) approaches 1 from the left, \( f(x) = 0 \).
- As \( x \) approaches 1 from the right, \( f(x) = -1 \).
- Therefore, \( f(x) \) is not continuous at \( x = 1 \).
5. **Differentiability**:
- Since \( f(x) \) is not continuous at \( x = 1 \), it cannot be differentiable there.
- However, \( f(x) \) is a constant function in the intervals \( (-1, 0) \) and \( (0, 1) \), which means it is differentiable in these intervals.
### Conclusion
Based on the analysis, we can summarize:
- \( f(x) \) is continuous at \( x = 0 \).
- \( f(x) \) is continuous in the interval \( (-1, 0) \).
- \( f(x) \) is not differentiable at \( x = 1 \).
- \( f(x) \) is differentiable in the interval \( (-1, 1) \).
### Final Answer:
- The function \( f(x) \) is continuous at \( x = 0 \) and in the interval \( (-1, 0) \), and differentiable in the interval \( (-1, 1) \).
